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Section 13.1

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2 Review right triangle trigonometry from Geometry and expand it to all the trigonometric functions Begin learning some of the Trigonometric identities

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Evaluate trigonometric functions of acute angles. Use fundamental trigonometric identities. Use a calculator to evaluate trigonometric functions. Use trigonometric functions to model and solve real-life problems.

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4 Trigonometry is based upon ratios of the sides of right triangles. The ratio of sides in triangles with the same angles is consistent. The size of the triangle does not matter because the triangles are similar (same shape different size).

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5 The six trigonometric functions of a right triangle, with an acute angle , are defined by ratios of two sides of the triangle. The sides of the right triangle are: the side opposite the acute angle , the side adjacent to the acute angle , and the hypotenuse of the right triangle. opp adj hyp θ

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6 The trigonometric functions are sine, cosine, tangent, cotangent, secant, and cosecant. opp adj hyp θ sin = cos = tan = csc = sec = cot = opp hyp adj hyp adj opp adj Note: sine and cosecant are reciprocals, cosine and secant are reciprocals, and tangent and cotangent are reciprocals.

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7 Another way to look at it… sin = 1/csc csc = 1/sin cos = 1/sec sec = 1/cos tan = 1/cot cot = 1/tan

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8 5 12

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9 Calculate the trigonometric functions for . The six trig ratios are 4 3 5 sin = tan = sec = cos = cot = csc = cos α = sin α = cot α =tan α = csc α = sec α = What is the relationship of α and θ? They are complementary (α = 90 – θ) Calculate the trigonometric functions for .

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10 Note : These functions of the complements are called cofunctions. Note sin = cos(90 ), for 0 < < 90 Note that and 90 are complementary angles. Side a is opposite θ and also adjacent to 90 ○ – θ. a hyp b θ 90 ○ – θ sin = and cos (90 ) =. So, sin = cos (90 ).

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11 Geometry of the 45-45-90 triangle Consider an isosceles right triangle with two sides of length 1. 1 1 45 The Pythagorean Theorem implies that the hypotenuse is of length.

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12 Calculate the trigonometric functions for a 45 angle. 1 1 45 csc 45 = = = opp hyp sec 45 = = = adj hyp cos 45 = = = hyp adj sin 45 = = = cot 45 = = = 1 opp adj tan 45 = = = 1 adj opp

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13 Calculate the trigonometric functions for a 30 angle. 1 2 30 csc 30 = = = 2 opp hyp sec 30 = = = adj hyp cos 30 = = hyp adj tan 30 = = = adj opp cot 30 = = = opp adj sin 30 = =

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14 Calculate the trigonometric functions for a 60 angle. 1 2 60 ○ csc 60 = = = opp hyp sec 60 = = = 2 adj hyp cos 60 = = hyp adj tan 60 = = = adj opp cot 60 = = = opp adj sin 60 = =

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15 SineCosineTangent 30 0 45 0 1 60 0

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* Find the value of x for the right triangle shown

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* Solve ∆ABC a= 4.48 and c = 13.8 a/13 = tan 19 c/13 = sec 19

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The angle you are given is the angle of elevation, which represents the angle from the horizontal upward to an object. For objects that lie below the horizontal, it is common to use the term angle of depression.

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A surveyor is standing 115 feet from the base of the Washington Monument. The surveyor measures the angle of elevation to the top of the monument as 78.3 . How tall is the Washington Monument? Figure 4.33

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where x = 115 and y is the height of the monument. So, the height of the Washington Monument is y = x tan 78.3 115(4.82882) 555 feet.

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30,000 ft

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